Problem: You have found the following ages (in years) of all 5 lizards at your local zoo: $ 1,\enspace 1,\enspace 2,\enspace 4,\enspace 1$ What is the average age of the lizards at your zoo? What is the variance? You may round your answers to the nearest tenth.
Solution: Because we have data for all 5 lizards at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $5$ ages and divide by $5$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5}} $ $ {\mu} = \dfrac{1 + 1 + 2 + 4 + 1}{{5}} = {1.8\text{ years old}} $ Find the squared deviations from the mean for each lizard. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $1$ year $-0.8$ years $0.64$ years $^2$ $1$ year $-0.8$ years $0.64$ years $^2$ $2$ years $0.2$ years $0.04$ years $^2$ $4$ years $2.2$ years $4.84$ years $^2$ $1$ year $-0.8$ years $0.64$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{0.64} + {0.64} + {0.04} + {4.84} + {0.64}} {{5}} $ $ {\sigma^2} = \dfrac{{6.8}}{{5}} = {1.36\text{ years}^2} $ The average lizard at the zoo is 1.8 years old. The population variance is 1.36 years $^2$.